I'm Chris, Thanks for allowing me to help you here on Just Answer.
I need to know some voltages
A good battery will have 12.4-12.6 volts. So getting a reading of 12.1 and then 12.0 means the battery is discharged quite a bit. Meaning it only has about 50% of the amps it claims to have. Since the voltage dropped on the light test this means the reserve capacity is also struggling since the battery has very low power. With a battery this low you should be seeing charging voltages more near 14-14.8. The voltages as they are are fine but not with a battery showing this voltage. In this case you first off need to get a long battery charge on it to get this battery back up to health and see why this alternator is not working up to par.
This chargeing system is computer controlled (PCM) so you need to get a good scan tool on it to see what the computer is commanding commpared to what the alternator is putting out. I feel your looking at a alternator which is just not a good one and cant handle the needed output.